Exceptional extensions of local fields and the Carlitz--Wan conjecture
Zhiguo Ding, Wei Xiong, Qifan Zhang
TL;DR
The paper develops a local-field analogue of exceptional covers and establishes that, for a finite separable extension of local fields $L/K$ with residue field $\mathbb{F}_q$, exceptionality forces $\gcd([L:K],q-1)=1$ and is tightly connected to ramification behavior. It builds a comprehensive theory of exceptional local-field extensions, including multiple equivalent characterizations and preservation under subextensions, then leverages this framework to provide three distinct proofs of the Carlitz–Wan conjecture (and its generalization) via global-to-local reductions and purely local, group-theoretic arguments. A central result ties ramification indices to the absence of certain Galois subextensions, yielding new proofs of a theorem of Guralnick–Müller and deepening the link between exceptional covers of curves and local-field ramification. Overall, the work unifies function-field phenomena with local-field ramification theory and introduces robust tools for analyzing exceptional maps in positive characteristic.
Abstract
For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one $k$ with $q^k\ge °(f)^4$. In this paper, we introduce the notion of an ``exceptional'' extension of local fields of any characteristic, and show that if $f(X)\in\F_q[X]$ is exceptional in the classical sense then the field extension $\F_q(X)/\F_q(f(X))$ yields an exceptional local field extension upon passing to the completion at a degree-$1$ place. We describe all exceptional local field extensions of degree coprime to the residue characteristic, determine the relationship between exceptionality of a local field extension and exceptionality of a subextension, and give various Galois-theoretic characterizations of exceptional local field extensions. As a consequence, we obtain three new proofs, using quite different tools, of a theorem of Guralnick and Müller about ramification indices in exceptional maps between curves over $\F_q$. This theorem generalizes a result of Lenstra which subsumes earlier conjectures of Carlitz and Wan.